El e c t ro n ic
Jo ur
n a l o
f P
r o
b a b il i t y
Vol. 14 (2009), Paper no. 84, pages 2418–2437. Journal URL
http://www.math.washington.edu/~ejpecp/
Generalised stable Fleming-Viot processes as flickering
random measures
Matthias Birkner
∗, Jochen Blath
†,
Abstract
We study some remarkable path-properties of generalised stable Fleming-Viot processes (includ-ing the so-called spatial Neveu superprocess), inspired by the notion of a “wander(includ-ing random measure” due to Dawson and Hochberg (1982). In particular, we make use of Donnelly and Kurtz’ (1999) modified lookdown construction to analyse their longterm scaling properties, ex-hibiting a rare natural example of a scaling family of probability laws converging in f.d.d. sense, but not weakly w.r.t. any of Skorohod’s topologies on path space. This phenomenon can be ex-plicitly described and intuitively understood in terms of “sparks”, leading to the concept of a “flickering random measure”.
In particular, this completes results of Fleischmann and Wachtel (2006) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process.
Key words: Generalised Fleming-Viot process, flickering random measure, measure-valued dif-fusion, lookdown construction, wandering random measure, Neveu superprocess, path proper-ties, tightness, Skorohod topology.
AMS 2000 Subject Classification:Primary 60G57; Secondary: 60G17.
Submitted to EJP on September 16, 2008, final version accepted November 2, 2009.
∗Institut für Mathematik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany. E-mail:
birkner@mathematik.uni-mainz.de
†Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. E-mail:
1
Introduction and statement of the main results
1.1 Classical and generalised Fleming-Viot processes
In 1979, Fleming and Viot introduced their now well-known probability-measure-valued stochastic process as a model for the distribution of allelic frequencies in a selectively neutral genetic popula-tion with mutapopula-tion (cf.[FV79]). More formally, they introduced a Markov process{Yδ0,∆
t ,t ≥0},
with values inM1(Rd)(denoting the probability measures onRd), such that for functionsF of the form
with∆the Laplace operator. The meaning of the superscripts in {Yδ0,∆
t ,t ≥0}will become clear
once we identify this process as a special case of a much larger class of processes.
It is well known (cf.[DH82]) that the classical Fleming-Viot process is dual toKingman’s coalescent, introduced in[K82], in the following sense (our description being rather informal). Fort≥0, if one takes a uniform sample ofnindividuals fromYδ0,∆
t and forgets about the respective spatial positions
of the nparticles, then their genealogical tree backwards in time can be viewed as a realisation of Kingman’s n-coalescent. That means, at each time t−s, wheres∈[0,t](hencebackwardsin time), the ancestral lineages of each particle merge at infinitesimal rate k2
, wherek∈ {2, . . . ,n}denotes the number of distinct lineages present at time t−s(−). This can be made rigorous, for example, using Donnelly and Kurtz’ lookdown construction ([DK96]), and spatial information may also be incorporated, see e.g.[Eth00], Section 1.12.
Since its introduction, the Fleming-Viot process received a great deal of attention from both geneti-cists and probabilists. One reason is that it is the natural limit of a large class of exchangeable population models with constant size and finite-variance reproduction mechanism, in particular the so-called Moran-model, and can be viewed as the infinite-dimensional analogue of the Wright-Fisher diffusion. See[Eth00]for a good overview.
More general limit population processes describing situations where, from time to time, a single individual produces a non-negligible fraction of the total population have been introduced in[DK99]
(see also[BLG03]for a different approach). We follow[BLG03]in calling such processesgeneralised Fleming-Viot processes. The limits of their dual genealogical processes have been classified in[Sa99],
[MS01]. See[BB09]for an overview. Generalised Fleming-Viot processes are probability measure valued Markov processes YΛ,∆α whose generator acts on functions F of the form (1.1) withφ
where
λn,k=
Z
[0,1]
xk−2(1−x)n−kΛ(d x), n≥k≥2, (1.3)
withΛa finite measure on[0, 1], and∆α=−(−∆)α/2is the fractional Laplacian of indexα∈(0, 2], see e.g.[Y65], Chapter IX.11, or[Fe66], Chapter IX.6, i.e., ∆α is the generator of the semigroup
(Pt(α))t≥0 of the d-dimensional standard symmetric stable process {B(tα),t ≥ 0} of index α. Note that for notational convenience, we denote by(Pt(2))t≥0 the semigroup ofd-dimensional Brownian motion with covariance matrix 2Id at time 1.
Remark 1.1. The special form (1.2) which the generalized Fleming-Viot generator takes when acting on functions of type (1.1) highlights its connection to the corresponding dual coalescent processes. Note that (1.2) has been derived e.g. in the proof of Theorem 3 in[BLG03], characterizing theΛ -Fleming-Viot process as solution to a well-posed martingale problem (which implies the strong Markov property); see also[DK99, Thm. 4.3]. For the ‘general form’ of the generator of theΛ-Fleming-Viot process and its construction as flow of bridges see Section 5.1 and (16) in[BLG03], or, alternatively, the explicit construction via particle systems in [DK99]or [BBM+09], where the latter reference also provides a classical construction via the Hille-Yosida Theorem.
We endowM1(Rd)with the topology of weak convergence, which we think of being induced by the ProhorovmetricdM1, defined forµ,ν∈ M1(R
d)by
dM1(µ,ν):=inf
ǫ >0 :µ(B)≤ν(Bǫ) +ǫfor all closedB⊂Rd , (1.4)
whereBǫis the usual openǫ-enlargement of the setB⊂Rd. It is well known thatdM
1is a complete
metric onM1(Rd), cf. e.g.,[EK86], Thm. 3.1.7 and Thm. 3.3.1. By[DK99, Theorem 3.2], the processes{YΛ,∆α
t ,t≥0}take values inD[0,∞)(M1(Rd)), the space of càdlàg paths, endowed with the usual Skorohod (J1-)topology (cf.[S56], or[Bi68], Chapter 3).
For a given Λ ∈ Mf([0, 1]), the rates λn,k describe the transitions of an exchangeable
partition-valued process{ΠΛt,t ≥ 0}, the so-called Λ-coalescent ([Pi99], [Sa99]). Indeed, for t ≥ 0, while
ΠΛt has n classes, any k-tuple merges to one at rate λn,k. A Λ-Fleming-Viot process is dual to
a Λ-coalescent (as shown in [DK99], pp. 195 and [BLG03]), similar to the duality between the classical Fleming-Viot process and Kingman’s coalescent established in[DH82]. Note that Kingman’s coalescent corresponds to the choiceΛ =δ0.
1.2 Generalised Fleming-Viot processes and infinitely divisible superprocesses
Fleischmann and Wachtel ([FW06]) have considered a probability measure valued process{Yt,t ≥
0} obtained by renormalising a spatial version of Neveu’s continuous mass branching process
{Xt,t ≥ 0} with underlying α-stable motion (as constructed e.g. in [FS04]via approximation or implicitly in [DK99]) with its total mass, i.e. 〈φ,Yt〉 = 〈φ,Xt〉/〈1,Xt〉, and have investigated its
long-time behaviour.
In [BBC+05], the relation between stable continuous-mass branching processes {Zt,t ≥ 0} and
Beta(2−β,β)-Fleming Viot processes, for β ∈ (0, 2], (with a “trivial” spatial motion) has been explored. Informally,Zt/〈1,Zt〉, time-changed with the inverse of
Z t
0
is a Beta(2−β,β)-Fleming Viot process. This can be viewed as an extension of Perkins’ classical disintegration theorem ([EM91], [Pe91]) to the stable case. It is in principle easy to include a spatial motion component, but note that then the corresponding Fleming-Viot process uses a time-inhomogeneous motion, namely anα-stable process time-changed by the inverse of (1.5). However, Neveu’s branching mechanism is stable of index β = 1, so that the time change induced by (1.5) becomes trivial. Thus we obtain
Proposition 1.2(Normalised spatial Neveu branching process as generalised Fleming-Viot process).
Under the above conditions, we have
{Xt/〈1,Xt〉,t≥0} d
={YU,∆α
t ,t≥0},
where U=Beta(1, 1)is the uniform distribution on[0, 1].
Note that in particular in this situation, the (randomly) renormalised process{Xt/〈1,Xt〉,t ≥0} is
itself a Markov process. In fact, as observed in [BBC+05], it is the only “superprocess” with this property. This observation was the starting point of our investigation.
By consideringF as in (1.1) withn=1 resp.n=2, it follows from the martingale problem for (1.2) that the first two moments of a generalisedΛ-Fleming-Viot process only depend on the underlying motion mechanism and the total massΛ([0, 1]).
Proposition 1.3(First and second moment measure). Let Y0=µ∈ Mf\{0}. Then,
E〈ϕ,YΛ,∆α t 〉
=
Z
Pt(α)ϕ(x)µ(d x), (1.6)
and for t1≤t2, writingρ:= Λ([0, 1]), andϕ,ϕ1,ϕ2∈Cb2,
E〈ϕ1,YΛ,∆α t1 〉〈ϕ2,Y
Λ,∆α
t2 〉
=
Z t1
0
Z
ρe−ρsPs(α) Pt(1−α)sϕ1P (α)
t2−sϕ2
(x)µ(d x)ds
+e−ρt1 Z
Pt(1α)ϕ1(x)µ(d x)
Z
Pt(2α)ϕ2(x)µ(d x). (1.7)
In particular, the first two moment measures agree with those of the classical Fleming-Viot process, which explains Proposition 3 in[FW06]. Note that in order to establish (1.6) and (1.7), one cannot apply the Laplace-transform method as in [FW06]since the branching property does not hold in general. A proof can be found in Section 3.
Note that a simple explanation can be given in terms of thedual coalescent processmentioned before. Indeed, in [BLG03] it is shown that generalised Fleming-Viot processes are moment dual to Λ -coalescents (see also [BB09]). Since the first two moments do not involve multiple coalescent events, theycannot “feel” the finer properties of the measure Λ. Of course, for moments greater than two, the moment formulae cannot be expected to agree.
1.3 Coherent wandering random measures
inRd and for eachǫ >0 a real-valued stationary “radius process”{Rǫ(t),t ≥0}and an a.s. finite T0, such that
Yδ0,∆
t Bx(t)(Rǫ(t))≥1−ǫ fort≥T0 a.s., (1.8) where Bx(r) is the closed ball of radius r around x ∈ Rd. One natural choice for {x(t),t ≥ 0}
is the “centre of mass process” x(t) = R x Yδ0,∆
t (d x), see [DH82], Equation 3.10. However, in
the context of the lookdown construction, a more convenient choice is x(t) =ξ1t, the location of the so-called “level-1 particle” (see Section 2). With this choice, an obvious extension of[DK96], Theorem 2.9, shows that anyYΛ,∆αis a coherent wandering random measure. If the process YΛ,∆α
has the compact support property, i.e.,almost surely,
supp YΛ,∆α t
is compact for allt,
this will also yieldcompact coherence, i.e. one can chooseǫ=0 in (1.8), see[DH82], Theorem 7.2.
It is interesting to see that generalised Fleming-Viot processes need not have the compact support property, even if the underlying motion is Brownian and the initial state has compact support.
Indeed, if the dualΛ-coalescent ΠΛ does not come down from infinity, i.e. if starting fromΠΛ0 = {{1},{2}, . . .}, the number of classes|ΠΛt|ofΠΛt is (a.s.) infinite for any t>0, then
supp YtΛ,∆
=Rd a.s. for anyt.
Recall that if the standardΛ-coalescent does not come down from infinity (a necessary and sufficient condition for this can be found in[Sc00]), it either has a positive fraction of singleton classes (so-called “dust”), or countably many families with strictly positive asymptotic mass adding up to one (so called “proper frequencies”), cf.[Pi99], Lemma 25. Using the pathwise embedding of the standard
Λ-coalescent in the Fleming-Viot process provided by the modified lookdown construction (see (2.7) below) we see that in the first case, the positive fraction of singletons contributes an α-heat flow component to YΛ,∆α
t , whereas in the latter case there are infinitely many independent families of
strictly positive mass, so that by the Borel-Cantelli Lemma any given open ball inRd will be charged almost surely.
Combining this with Proposition 1.2, we recover Proposition 14 of[FS04]on the instant propagation of the spatial Neveu branching process.
Remark 1.4. Observe that for continuous test functionsϕ with compact support,
td/αE
h
〈ϕ,YΛ,∆α t 〉
i
→ p(1α)(0)
Z
ϕ(x)d x ast → ∞, (1.9)
where p(tα)(x) is the transition density of{Bt(α),t ≥ 0}. This is essentially Corollary 6 of[FW06], which was formulated for Λ = U only. In the subsequent Remark 7, Fleischmann and Wachtel ask about convergence of td/α〈ϕ,YU,∆α〉. With the lookdown construction in mind, (1.9) can be
understood as follows: without loss of generality assume that ϕ has support in the unit ball, put Ct := 〈ϕ,YΛ,∆α
t 〉. Consider the empirical process {Y
Λ,∆α
t ,t ≥ 0} together with {ξ1t,t ≥ 0}, the
position of the level-1 particle. ThenYΛ,∆α
t (· −ξ1t)converges to some stationary distribution (again,
substantial, whereas otherwise it is essentially zero. The terms balance exactly, so that the lefthand side of (1.9) converges, but in fact as{B(tα),t≥0}is not positive recurrent,Ct converges to zero in distribution (and even a.s. ifα <d, i.e. ifξ1t is transient).
1.4 Main results: Longterm-scaling, existence of sparks, and flickering random measures
The long-time behaviour of a generalised Fleming-Viot process reflects the interplay between motion and resampling mechanism. If one attempts to capture this via a space-time rescaling, the scaling will be dictated by the underlying (stable) motion process:LetΛ∈ Mf([0, 1])such thatΛ([0, 1))>
0and define the rescaled process{YΛ,∆α
t [k],t≥0}via
φ,YΛ,∆α t [k]
:=
φ(·/k1/α),YΛ,∆α kt
, (1.10)
forφ∈bB(Rd)andt≥0. LetB(α), forα∈(0, 2], be the standard symmetric stable process of index
α, starting from B0(α)=0. With these definitions, one readily expects the following convergence of finite-dimensional distributions (f.d.d.) to hold:
Proposition 1.5 (Longterm-Scaling). For each finite collection of time-points 0≤ t1 <· · ·< tn, we have
YΛ,∆α
t1 [k], . . . ,Y Λ,∆α
tn [k]
⇒
δB(α)
t1
, . . . ,δB(α)
tn
as k→ ∞. (1.11)
Note that for the classical {Yδ0,∆
t ,t ≥ 0}, this is essentially Theorem 8.1 in [DH82]. Combining
Proposition 1.2 and Proposition 1.5, we recover Part (a) of Theorem 1 in[FW06].
In addition to f.d.d.-convergence, Part (b) of Theorem 1 in [FW06] provides weak convergence on D[0,∞)(M1(Rd))if the underlying motion of the spatial Neveu process (i.e. Λ = U) is Brown-ian. However, the question whether this holds in general seems to be inaccessible to the Laplace-transform and moment-based methods of[FW06], and therefore had been left open.
Rather surprisingly, it turns out that pathwise convergence doesnothold ifα <2, and that, with the help of Donnelly and Kurtz’ modified lookdown construction ([DK99]), it is possible to understand explicitly “what goes wrong”. To this end, we introduce the concept of “sparks” and of a family of “flickering random measures”.
Definition 1.6 (Sparks). Consider a path ω = {ωt,t ≥ 0} in D[0,∞)(M1(Rd)). We say that ω exhibits anǫ-δ-spark(on the interval[0,T]) if there exist time points0<t1<t2< t3≤T such that t3−t1≤δ
dM1(ωt1,ωt3)≤ǫ, dM1(ωt1,ωt2)≥2ǫ and dM1(ωt2,ωt3)≥2ǫ, (1.12)
where dM1 denotes the metric (1.4) onM1(Rd).
Definition 1.7(Flickering random measures). Let{Z[k],k∈N}be a family of measure-valued pro-cesses on D[0,∞)(M1(Rd)). If there exists anǫ >0and a sequenceδk↓0, such that
lim inf
k→∞
PZ[k]exhibits anǫ-δk-spark in[0,T] >0,
The space-time scaling family of many generalised Fleming-Viot processes satisfies this definition:
Lemma 1.8 (Generalised Fleming-Viot processes as flickering random measures). If α < 2 and
Λ((0, 1))>0, there existsǫ >0such that
lim inf
k→∞
PYΛ,∆α[k]exhibits anǫ-(1/k)-spark in[0,T] >0.
Hence, the scaling family{YΛ,∆α[k],k ≥1} is a family of“flickering random measures” withδ k =
1/k,k∈N .
We will see below that the behaviour of{YΛ,∆α[k], leading to anǫ–1/k-spark described by condition
(1.12) typically arises as follows: At timest1 andt3, YΛ,∆α[k]is (almost) concentrated in a small
ball with (random) centre x, say. At time t2, suddenly a fractionǫ of the total mass appears in a remote ball with centre y, where|x−y| ≥1, and vanishes almost instantaneously, i.e., by time t3.
Technically, we see that Lemma 1.8 shows that the modulus of continuityw′(·,δ,T)of the processes YΛ,∆α[k], see (3.5) below, does not become small as δ → 0, contradicting relative compactness
of distributions on D[0,∞)(M1(Rd)). Intuitively, at each infinitesimal “spark”, a limiting process is neither left- nor right-continuous. We will see below how this intuition can be made precise in the framework of the (modified) lookdown construction.
The situation is different if Λ = cδ0 for some c > 0 and α <2. Here, each Ycδ0,∆α[k] a.s. has continuous paths, so that any limit in Skorohod’s J1-topology would necessarily have continuous paths. However, the f.d.d. limit{δB(α)
t
,t ≥0}has no continuous modification. Intuitively, there is no “flickering”, but an “afterglow” effect: >From time to time, a very fertile “infinitesimal” particle jumps some distance, and then founds an extremely large family, so that the population quickly becomes essentially a Dirac measure at this point, while at the same time the rest of the population (continuously) “fades away”.
To complete the picture, we are finally able to provide the full classification of the scaling behaviour of generalisedα-stable Fleming-Viot processes.
Theorem 1.9(Convergence on path space). IfΛ([0, 1))>0,(1.11) holds weakly on D[0,∞)(M1(Rd)) if and only ifα=2.
Remark 1.10 (Other Skorohod topologies). Note that the above-mentioned “afterglow”-phenomenon in the case Λ = cδ0 and α < 2 fits well to Skorohod’s M1-topology (see [S56], Definition 2.2.5), which is tailor-made to establish convergence in situations in which a discon-tinuous process is approximated by a family of condiscon-tinuous processes. However, in the situation of Lemma 1.8, Condition (1.12) implies that the distributions of the processes YΛ,∆α[k] cannot
converge with respect to any of the topologies considered in[S56]. We are not aware of a reason-able topologyT onD[0,∞)(M1(Rd))such that (the distribution of){YΛ,∆α[k](t)}converges weakly towards (the distribution of){δB(α)
t }on(D[0,∞)(M1(
Rd)),T).
Remark 1.11(The case of star-shaped genealogies). Note that the caseΛ =cδ1,c>0, has been ex-cluded from the setup of this subsection. Here, occasionally (i.e. with ratec), the whole population always jumps to the position of the level-1-particle, producing a star-shaped genealogy. Inbetween, theα-heat flow acts so that the process is continuous between consecutive jumps. Hence, it is clear thatYcδ1,∆α[k]converges weakly to the stable unit motionδ
2
Donnelly and Kurtz’ lookdown construction
2.1 A countable representation for generalised Fleming-Viot processes
We consider a countably infinite system of individuals, each particle being identified by a levelj∈N. We equip the levels with typesξtj inRd, j∈N. Initially, we require the typesξ0= (ξ0j)j∈N to be an i.i.d. vector (in particular exchangeable), so that
lim
N→∞ 1 N
N
X
j=1
δξj
0
=µ,
for some probability measure µ ∈ M1(Rd), which will be the initial condition of the generalised Fleming-Viot process constructed below via (2.6). The point is that the construction will preserve exchangeability.
There are two “sets of ingredients” for the reproduction mechanism of these particles, one corre-sponding to the “finite variance” part Λ({0}), and the other to the “extreme reproductive events” described byΛ0= Λ−Λ({0})δ0. Restricted to the firstN levels, the dynamics is that of a very par-ticular permutation of a generalised Moran model with the property that always the particle with the highest level is the next to die.
For the first part, let {Li j(t),t ≥ 0}, 1 ≤ i < j < ∞, be independent Poisson processes with rate Λ({0}). Intuitively, at jump times of Li j, the particle at level j “looks down” to level iand copies
the type from there, corresponding to a single birth event in a(n approximating) Moran model. Let ∆Li j(t) = Li j(t)−Li j(t−). At jump times, types on levels above j are shifted accordingly, in formulas
ξkt =
ξkt−, if k< j,
ξit−, if k= j,
ξk−1
t− , if k> j,
(2.1)
if ∆Li j(t) = 1. This mechanism is well defined because for each k, there are only finitely many processes Li j,i< j≤kat whose jump timesξkhas to be modified.
For the second part, which corresponds to multiple birth events, let n be a Poisson point process onR+×(0, 1]×[0, 1]N
with intensity measured t⊗r−2Λ0(d r)⊗(du)⊗N. Note that for almost all realisations{(ti,yi,(ui j))}ofn, we have
X
i:ti≤t
yi2<∞ for allt ≥0. (2.2)
The jump times ti in our point configuration ncorrespond to reproduction events. Define for l ∈
N,l≥2 andJ⊂ {1, . . . ,l}with|J| ≥2,
LJl(t):= X i:ti≤t
Y
j∈J
1ui j≤yi Y
j∈{1,...,l}−J
1ui j>yi. (2.3)
LlJ(t)counts how many times, among the levels in {1, . . . ,l}, exactly those in J were involved in a birth event up to timet. Note that for any configurationnsatisfying (2.2), since|J| ≥2, we have
ELl
J(t)
n|[0,t]×(0,1]= X
i:ti≤t
yi|J|(1−yi)l−|J|≤ X i:ti≤t
new particle
Figure 1: Relabelling after a birth event involving levels 2, 3 and 6.
so that LJl(t)is a.s. finite.
Intuitively, at a jumpti, each level performs a “uniform coin toss”, and all the levels j withui j ≤ yi
participate in this birth event. Each participating level adopts the type of the smallest level involved. All the other individuals are shifted upwards accordingly, keeping their original order with respect to their levels (see Figure 1). More formally, if t = ti is a jump time and j is the smallest level
involved, i.e.ui j ≤yi anduik> yi fork< j, we put
is theσ-algebra describing all “genealogical events” between timesuandv.
So far, we have only treated the reproductive mechanism of the particle system. In-between repro-duction events, all the levels follow independentα-stable motions. For a rigorous formulation, all three mechanisms together can be cast into a suitable countable system of stochastic differential equations driven by Poisson processes andα-stable processes, see[DK99], Section 6.
exists almost surely onD[0,∞)(M1(Rd)), and{Zt,t≥0}is the Markov process with generator (1.2) and initial conditionZ0=µ, see[DK99], Theorem 3.2.
2.2 Pathwise embedding ofΛ-coalescents inΛ-Fleming-Viot processes
Note that for each t > 0 and s ≤ t, the modified lookdown construction encodes the ancestral partition of the levels at timet with respect to the ancestors at timesbefore t via
Nit(s) =level of leveli’s ancestor at timet−s.
For fixed t, the vector-valued process{Nit(s):i∈N}0≤s≤t satisfies an “obvious” system of Poisson-process driven stochastic differential equations, see[DK99], p. 195, (note that we have indulged in a time re-parametrisation), and the partition-valued process defined by
{i:Nit(s) =j},j=1, 2, . . . (2.7)
is a standardΛ-coalescent with time interval [0,t]. This implies in particular by Kingman’s theory of exchangeable partitions (see[K82], or, e.g.,[Pi06]for an introduction), that the empirical family sizes
Atj(s):= lim
n→∞ 1 n
n
X
i=1
1{Nit(s)=j} (2.8)
exist a.s. in[0, 1]for each j ands≤t, describing the relative frequency at time tof descendants of the particle at level jat time t−s.
3
Proofs
Fixµ∈ M1(Rd)as the initial condition of the un-scaled process YΛ,∆α. We begin with the useful
observation that, due to the scaling properties of the underlying motion process, for each k, the process{Yt(k),t≥0}, defined by
Yt(k)=YkΛ,∆α
t , t≥0, (3.1)
starting from the image measure ofµunder x7→x/k1/α, has the same distribution as{YΛ,∆α
t [k],t≥
0}defined in (1.10). It will be convenient to work in the following with a version ofY(k)which is obtained from a lookdown construction with “parameter”kΛ, in particular, we have
Yt(k)= lim
n→∞ 1 N
N
X
i=1
δξi
t, t≥0.
Note that the familyξi, i∈N, used to constructY(k)depends (implicitly) on k, but for the sake of readability, we suppress this in our notation.
3.1 Proof of Proposition 1.5
Instead of following the arguments of [FW06, Lemma 19](which only make use of the formulas for the first two moments and thus in view of Prop. 1.3 could be adapted as well), we give a proof which is directly based on the lookdown-construction. Indeed, we show that the path of the unit mass {δB(α)
t ,t ≥ 0} can be viewed as the trail of the level-1-particle. To this end, we first show
convergence in law of the one-dimensional distributions, i.e.
YΛ,∆α
t [k]⇒δBt(α), ask→ ∞, t≥0.
Since the motion of the level-1-particle{ξ1t,t≥0}is a symmetricα-stable process,i.e.B(tα)=dξ1t, it sufficesby the special form of the limit variableto check that
lim
t and radiusǫ. The latterwill be implied by
In order to check this, letΦǫ be a “mollified” (continuous) indicator ofBξ1 t(ǫ)
c, and note, by
domi-nated convergence, that for anyδ >0
E〈Φǫ,Y(k) Conditioning onGt−δ,t, the genealogical information as defined in (2.5), we estimate the first term as follows:
3.2 Proof of Theorem 1.9
In the caseα=2, using Proposition 1.3, tightness on the space D[0,∞)(M1(Rd))can be proved by inspection, literally tracing through the corresponding arguments of [FW06], Lemma 20 and 21 (note that even though Equations (133)–(137) in [FW06] estimate a fourth moment, this refers only to an increment of ad-dimensional Brownian motion).
For the caseα <2 andΛ((0, 1))>0, let us recall the following classical characterisation of relative compactness in D[0,∞)(M1(Rd)), cf. e.g.[Bi68], Theorem 15.2.
Theorem 3.1(Relative compactness on path space). Let{Yk}be a sequence of processes taking values in D[0,∞)(M1(Rd)). Then{Yk}is relatively compact if and only if the following two conditions hold:
• For everyǫ >0and every (rational) t≥0, there exists a compact setγǫ,t⊂ M1(Rd), such that lim inf
k→∞ P¦Yk
t ∈γǫ,t
©
≥1−ǫ.
• For everyǫ >0and T >0, there existsδ >0, such that
lim sup
k→∞
P¦w′(Yk,δ,T)≥ǫ©≤ǫ, (3.4)
where
w′(y,δ,T) =inf {ti}
max
i s,t∈[supt
i−1,ti)
d(y(s),y(t)), (3.5)
and{ti}ranges over all finite partitions of[0,T]such that ti−ti−1> δfor all i.
By Lemma 1.8, there is anǫ >0 such that fork0∈Nandδ >1/k0
P¦w′(YΛ,δα[k],δ,T)≥ǫ©≥PYΛ,∆α[k]exhibits anǫ-(1/k)-spark on[0,T]
is bounded away from 0 uniformly ink≥k0.
Finally, in the case α < 2 and Λ = cδ0 for some c > 0, note that, due to the absence of macro-scopic birth events, eachYcδ0,∆α[k]a.s. has continuous paths (formally, this follows e.g. from
The-orem 4.7.2 in[Da93]and the standard disintegration result, see[EM91], [Pe91]). Let µk denote
the distribution ofYcδ0,∆α[k]. Since the set of continuous pathsC :=C
[0,∞)(M1(Rd))is closed in Skorohod’sJ1-topology, weak convergence onD[0,∞)(M1(Rd))to some distributionµwould imply that
1=lim sup
k→∞
µk(C)≤µ(C),
by the Portmanteau Theorem. However, the f.d.d. limit{δB(α)
t ,t≥0}has no continuous modification
forα <2, and we arrive at a contradiction.
3.3 Proof of Lemma 1.8
vicinity of the location of the level-1 particle. A “spark” arises if the level-2 particle jumps to a remote position and shortly afterwards participates in an extreme reproduction event involving a positive fraction of the current population, but not the level-1 particle. In this situation, a new atom appears in the support of Y(k), which is then removed quickly, since mass is attracted rapidly towards the position of the level-1 particle. Note that corresponding phenomena will occur on any level j≥2.
First, we collect some useful notation. Without loss of generality assume T = 1. The following choices for the constantsδ2 andǫare justified by Lemma 4.1 and Lemma 4.2 from the Appendix. Indeed, chooseδ1 ∈(0, 1) withΛ((δ1, 1))>0, and then ǫ= ǫ(δ1)> 0 such that for any y ∈Rd
(with the usual convention inf;= +∞) be the backwards time to the most recent common ancestor of the particles at level jand at level 1 at time t, and let
be the event that in the time interval(s,t], no lookdown event involving both levels 1 and 2 occurs. Furthermore, note that since symmetricα-stable processes do not have fixed times of discontinuity,
lim
i that at timebimost of the population
(including the level-2 particle) is sufficiently closely related to the level-1 particle and has not moved too far away in space, more precisely, recalling (2.8),
• Within the time-interval (ci,di], the event Ci(k) requires that the level-2 particle does not travel very far, and that there is a lookdown event involving a sufficiently large fraction of the population, butnotthe level-1 particle:
Ci(k):=Hci,di
• Finally, letDi(k)be the event that most of the mass returns to the location of the level-1 particle, and stays there (which essentially is the same behaviour as within(ai,bi]), namely,
Di(k):=
Now let us introduce a family ofσ-algebras containing our ingredients: RecallGu,v from (2.5) and letHi(k)be theσ-algebra generated byGa
Indeed,Oi(k) guarantees that the level-1-particle did not move more thanδ2/2 units away during
Hence, an application of (3.7) using the observation that|ξ1
andCi(k)that the level-2-particle is sufficiently far away from the location of the level-1-particle in
(bi,ci], stays away more than 2 units during(ci,di]and at some timeτ∈(ci,di]it accumulates a proportion of mass at leastδ1. Hence,
YΛ,∆α
τ [k] (Bξ1
τ(2)) c
≥δ1,
and we know from above that under the assumptions as before (again, givenOi(k)),
YΛ,∆α
Finally,D(ik) guarantees that most of the mass returns to the close vicinity of the level-1-particle at timeai+1, which is, due toB
i , more than 2 units away from the position of the level-2-particle at timeτ, but still close to its own position at timeτ. Hence, as before,
YΛ,∆α Using (3.6) again, we finally arrive at
dM1(Y
Λ,∆α
τ [k],Y
Λ,∆α
ai+1 [k])>2ǫ. (3.13)
Recalling Definition 1.6, we see that (3.11), (3.12) and (3.13) guarantee the existence of an ǫ
-(1/k)-spark in(ai,ai+1]. It remains to show that
inf
which yields the claim. In order to verify (3.14), note thatby (3.9), for alli<k,
lim
and that for fixedk, the family of events
is independent (by the usual independence properties of the driving Poisson processes and the in-crements of the motion processes for each level). Moreover,
∀k,i<k: P E(k) erties and tail behaviour of symmetricα-stable processes, see e.g.[Be96, Chapter VIII],
P B(k)
The probability of the genealogical components ofEi(k)does not converge to zero, since smaller time intervals (of length 1/k) are compensated by higher resampling intensities due to the time rescaling (proportional to k) in each step. Finally, the probability of the remaining requirements (upper bounds on the distance that the level-1-particle or the level-2-particle are allowed to travel) do not converge to zero either by (3.9) or the fact that these upper bounds are fixed values, independent ofk.
3.4 Proof of Proposition 1.3
The proof is a straightforward modification of arguments in[Eth00], Prop. 2.27. Indeed, it is easy to read off the function-valued dual process from (1.2), which is an obvious extension of [Da93], Section 5.6(see[BBM+09, Section 5.2]for an explicit description in the context ofΞ-Fleming-Viot processes, which containsΛ-Fleming-Viot processes as a special case).
For the first moment, the function valued dual consists of one component only, to which theα-heat flow is applied.This gives, for t>0,
E〈ϕ,YΛ,∆α t 〉
=
Z
Pt(α)ϕ(x)µ(d x). (3.16)
Note that this can easily be extended to distinct time points t1,t2 using the tower property and the
s is an independent copy ofY
Λ,∆α centered in y with radius r. We collect two technical lemmas.
Lemma 4.2. Pick some ǫ ∈ (0, 1/2). Then, there exists a constant δ˜(ǫ) > 0, such that for each 0< δ <δ˜and each x,x′with|x−x′| ≤δ, we have that for any two measuresµ,µ′∈ M1(Rd)
µ(Bx(δ))≥1−δ
2 and µ
′(B
x′(δ))≥1−
δ
2 (4.3)
implies
dM1(µ,µ ′)≤ǫ.
Proof. LetA⊂Rd be closed. Letδ <δ˜:=ǫ/4. We consider two cases. First, assumeA∩Bx(δ)6=;. Then,Bx(δ),Bx′(δ)⊂Aǫ. Hence,µ′(Aǫ) +ǫ≥1−δ/2+ǫ≥1 and the result holds.
In the second case, i.e.A∩Bx(δ) =;, we haveµ(A)< δ/2< ǫ, and the result holds, too.
Acknowledgement We would like to thank an anonymous referee for her/his careful reading and helpful comments.
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